Week #1 The Exponential and Logarithm Functions Section 1.2

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1 Week #1 The Exponential and Logarithm Functions Section 1.2 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. SUGGESTED PROBLEMS The functions in Exercises 1-4 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. 1. P = 5(1.07) t The initial quantity is 5. The growth rate is 7%, and is continuous. 5. A town has a population of 1000 people at time t = 0. In each of the following cases, write a formula for the population, P, of the town as a function of year t. (a) The population increases by 50 people a year. (b) The population increases by 5% a year. (a) P(t) = t (b) P(t) = 1000(1.05) t In Exercises 7-10, decide whether the graph is concave up, concave down, or neither. 7. Concave up. 8. Concave down. 9. Not concave (straight line) 1

2 10. Concave up. 11. Identify the x-intervals on which the function graphed in Figure 1.21 is: (a) Increasing and concave up (b) Increasing and concave down (c) Decreasing and concave up (d) Decreasing and concave down Figure 1.21 Problems (a) D-E, H-I (b) A-B, E-F (c) C-D, G-H (d) B-C, F-G 33. Estimate graphically the doubling time of the exponentially growing population shown in Figure Check that the doubling time is independent of where you start on the graph. Show algebraically that if P = P 0 a t doubles between time t and time t = d, then d is the same number for any t. Figure 1.22 To determine the doubling time graphically, it helps to pick two convenient points, say an interval on which the population has clearly doubled. For example, at t 4, P = 20,000, 2

3 and at t = 6, P = 40,000, or double the population at time t 4. If, for the purposes of the question, we estimate t = 3.8 to be the point when P = 20,000, this gives us a doubling time of d = = 2.2 years. We can check the doubling time by looking 2.2 years after t = 6, and see if the population has doubled up to 80, 000. Sure enough, t = = 8.2 has a population of 80, 000. Showing that the doubling time is consistent for any time interval for an exponential function requires looking means showing that P(t + d) = 2P(t) for any t, given that P(d) = 2P(0) (going from t = 0 to t = d doubles the population.). Since P(t) = P 0 e kt in general, this means that P(d) = P 0 e kd = 2P 0, or simply e kd = 2. Using the formula definition, P(t + d) = P 0 e kt+d = P 0 e kt e kd but e kd = 2, from earlier = P 0 e kt (2) = 2P(t) so the population has doubled after d years, regardless of the starting time t. QUIZ PREPARATION PROBLEMS 13. A photocopy machine can reduce copies to 80% of their original size. By copying an already reduced copy, further reductions can be made. (a) If a page is reduced to 80enlargement is needed to return it to its original size? (b) Estimate the number of times in succession that a page must be copied to make the final copy less than 15 (a) We have or Reduced Size = (0.80) Original Size Original Size = 1 Reduced Size 0.80 = 1.25 Reduced Size so the copy must be enlarged by a factor 1.25, or 125% to return it to its original size. (b) If a page is copied n times, then it has been reduced by (0.80) n. To get down to 0.80 n = 0.15, you can use logs, or trial and error to discover that = , so it would take 9 shrinking copies to reduce the image to less that 15% of its original size. 3

4 35. Each of the functions g, h, k in Table 1.7 is increasing, but each increases in a different way. Which of the graphs in Figure 1.23 best fits each function? Table 1.7 t g(t) h(t) k(t) Figure 1.23 (a) This is a linear function. In the table, for which the t values are evenly spaced, the y values should also be evenly spaced. This is only true of k(t), where each t gives rise to k of 0.3. (b) This function is increasing, and concave down. This means that the y increases are getting smaller as t increases. This happens for h(t). (c) This function is concave up and increasing, so the y increases should be getting larger as t increases. This happens for g(t). 36. (a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions. x f(x) g(x) h(x) (a) Since the x values are evenly spaced, a linear function would also have evenly spaced y values. h(x) satisfies this, as h = 3 for each step in x. Since the function goes through the point (0,31), and we computed the slope as h/ t = 3, we get the formula h(x) = 3x + 31 (b) Since the x values are evenly spaced, an exponential function would have a common multiplier for each increase in x. This occurs only for g(x), with the multiplier being 24/16 = 36/24 =... = 81/54 = 3/2. Since the point (0, 36) is part of g(x), the formula for g(x) would be 4

5 ( ) 3 t g(x) =